Bifurcations and safe regions in open Hamiltonians

Connecting Symmetric and Asymmetric Families of Periodic Orbits in Squared Symmetric Hamiltonians

“…The galactic potential 27,28 (n ¼ 1; ¼ À1; ¼ 0) with four escapes, that has been object of studies by some authors, and it has been observed to posses safe regular regions 29 for energies above the escape energy V ðx; yÞ ¼ 1 2 ðx 2 þ y 2 Þ À x 2 y 2 :…”

Breaking the limits: The Taylor series method

“…The normal modes behave as the organizing families of periodic orbits as all the other families tend to them and also configure the exit basin regions for large values of the energy. This system was studied in [33] and not only the use of the ETSM was quite useful for the OFLI2, but also in a precise integration for the computation of the escape basins and in the computation of the skeleton of symmetric periodic orbits. Also, in [33], several ''safe regions'' were located inside the escape sea (bounded regular regions inside the escape sea) that can lead to practical applications.…”

On the integration of an axially symmetric galaxy model

“…Over the last decades, the subject of escaping particles from an open dynamical system has been an active field of research. [1][2][3][4][5][6][7][8][9][10][11][12][13] Contopoulos 6 explored this phenomenon in Hamiltonian systems representing the central part of perturbed galaxies with quadruple symmetry and four openings to infinity, founding that the openings of the potential well are bridged by unstable periodic orbits. The asymptotic curves of these orbits govern the escape to infinity from the potential well.…”